NOVEL METRICS FOR LEARNING: LONG TERM RETENTION AS A SOCIO-ECONOMICINDEPENDENT AND SCHOOL COMPLETION AS A DEPENDENT METRIC.
VETURY SITARAMAM* and PALLAVI VETURY-IYER
"Vetury",# 18, S.No. 245, Anant Coop. Hsg. Society, Aundh, Pune - 411 067, India,
*
Corresponding author, [email protected].
Abstract
Efficacy of a system of education depends on whether it reaches the entire population and whether its
quality is acceptable, independent of the factors that limit its spread. We derive here novel metrics for
each. The triad of students, teachers and curriculum could be assessed by measuring long term retention
in school children by a recall method, which was independent of socioeconomic status, school grades
and gender but depends on the subject. The test also revealed that those who are not good at cramming
could exhibit good retention though the conventional tests fail to identify them. The socio–economic
dependence is seen specifically with school dropouts, where parental income plays a decisive role. This
economic influence on education follows a hierarchy of consumption wherein education itself is a
commodity of variable preference in the hierarchy of commodities. It is possible to extend the current
methodology to encompass the teachers and curriculum also into a quantifiable metric that helps render
examinations as an aid to learning rather than as a threat as commonly perceived.
Key words: STEM education, examinations, rote learning, cramming, econometrics, dropouts, income,
commodities, hierarchy of consumption, hyperbolic function.
1
Introduction
Raison d’être of education is in its social context – living in society as a contributor (to
‘perform’) and sharing its value system (to ‘belong’) [1-3]. A serious lacuna persists in evolving
appropriate frameworks for pedagogy in a just society: the act of learning per se could not be evaluated
independent of the socio-economic influences, thus far [4]. A major rethink on the structuring of
pedagogy in terms of teaching and evaluation practices are the need of the hour in the growing
aspiration for an effective and universal STEM education [5]. Students, teachers and curriculum form a
triad in the context of learning, requiring interdependent methods of evaluation, necessarily quantitative,
as it behooves STEM education itself [6], which enhances rigour and does not deemphasize learning in
humanities as well. The nub of the matter is in that the current methods of evaluation in education
reduce the individual performance to a set of scores or grades, much debated for their insight and
comprehensiveness [7]. The triad itself remains barely visible in current evaluation practices.
Three questions thereby arise when we examine the state of education and assessment in schools,
which relate to our assumptions as much as to our practices [8, 9]. Firstly there is a universal mistrust of
examination marks/grades which largely determine the future engagements of the students and their life
course. These examinations are primarily time-bound, content-bound (i.e., depending on the grade) and
largely age-bound, which ensure cramming as a necessity with little choice. Secondly, there is an
aversion to the so-called rote learning, which tends to be confused with cramming, while the former
should really reflect memorizing without any association/context in the strict sense of Ebbinghaus [10].
He coined meaningless triplet words and found that remembering these discrete, context-independent
particulate bits of information decays exponentially with time and the recall could be reinforced by
various stratagems. Cramming is a direct sequel to the practices of evaluation and not intrinsic to
learning. Both require some degree of comprehension and the ability to reproduce and to use the
concepts in the given context and yet learning requires that it is retained, the text and the context, with
clarity [11]. The third and yet the most important aspect of education is that an evaluation of teaching
and learning should transcend the socioeconomic barriers and aid and assess the child’s actual
intellectual potential and growth, independent of circumstances[12]. Any method that makes a clear
distinction is a significant step forward. Social justice must prevail.
These questions persist on the field of pedagogy largely due to deep methodological limitations,
which they share with social disciplines including psychology and economics. Headcount approaches, at
heart no more than anecdotal in their outlook and far from incisive, dominate as much as prescriptive
approaches bereft of quantitation for their efficacy. It would be preferable to examine issues of
pedagogy empirically and to let the conclusions decide the models for further testing since models need
to be both testable and falsifiable. This study initially probes in detail long term retention in a cross
section of students in an English medium school in urban India. Education remains one of the
commodities that a family spends as out of pocket expenses with marked differences in the quantum and
preference over other commodities and between rural and urban populations. The latter part models the
socio-economic aspects, manifest in dropout rates across the world, best documented in the published
Brazilian data [13,14]. Formal theoretical models that emerge uniquely permit future policy
considerations and quantitative monitoring by detailed trends rather than by head count approaches.
In this study, we modified the nature of the examination and measured the extent of the ‘failure’
of the system, as related to retention by students. These results demonstrate, in a cross section of school
children, a novel metric for the rate of fall in retention that is independent of school grades /
socioeconomic status / gender but solely depends on the content taught. It helps identify, among the
lesser scorers in school tests, those with better retention/comprehension, though less able to cram for
tests [15,16]. Since the students do vary in their ability to learn as much the teachers in their ability to
2
teach, the new metric of the fall in retention described here focuses uniquely on the need to determine
and distinguish quantitatively between the limits to learning and the limitations in imparting learning, a
prerequisite for an effective STEM education [cf. 6] of global concern and hence this communication.
This work was necessitated by the observed deterioration of education in India as elsewhere in the
world. The remedial changes wherever attempted were never with a metric for their efficacy or even
negative impact.
Theoretical considerations:
A. Retention of the past learning in a school
Since the conventional examinations in schools actually incentivize cramming, we tested the 9th
grade students for their retention from grade 1 to even grade 10 (as a control for the grade they were yet
to attend) without prior notice. The multiple choice tests were based on what they were actually taught
and the questions set by the teachers who taught these subjects. Fig. 1 summarizes the possible
outcomes…of the ability of the 9th grade students to recall what they learnt in the previous years, i.e.,
retention. Had the learning been perfect, the ideal representation would be invariant over the years as in
AB (~90% as in Model I, actually seen with students with conventional examinations and selected for
higher education in national tests [cf.16]). Had the learning been strictly by rote, it would be
increasingly more difficult to recall from the previous years and the results would approximate rote
learning and memory measured empirically as with Ebbinghaus [cf.10], i.e., Model II. Though his model
was based on time dependence of the limiting case of discrete, disconnected words without meaning,
various reinforcements for better memorizing such as repetition would tend to enhance recall, i.e., the
angle β to 900 approximating Model I. At the population level, precise relationships such as exponential
decay etc., would not be discernible beyond just trends, since the data would be strongly influenced by
distributions. The third alternative, in principle, would represent a disturbing element in assessing
education: the actual retention declines (α tending to zero) over years relentlessly (Model III). The
anticipation was that the students should not be able to answer the 10th grade questions, set as a logical
control since they are yet to begin to attend the grade.
A
B
90-α
Score →
α
90β
β
-x
C
9
1
10
+x
Grade →
Fig 1: Possible outcomes of retrospective testing of grade 9 students tested up to grade 10 after completion of their grade 9
year-end examinations as a linear approximation. A-B represents a perfect score of ~100 for all the years up to 9 and goes
down to 0 at grade 10 which the students are yet to begin (Model I). If the education were entirely by disconnected rote
learning, the Ebbinghaus model would predict B-(-x), where β follows the memorized content which disappears progressively
towards earlier grades (Model II). At best, variants of Ebbinghaus model on memory/retention would only predict that β
tends to zero at no retention or a right angle at full retention (Model I). However, if retention/learning is compromised in
education, the performance would deteriorate (α tending to zero) with successive years (A-C) (Model III). Note that one
would expect B or C to drop to near zero for Grade 10 component of the test. If it were to merely follow the regression line of
A-C, i.e. C-(+x)), it would represent a major anomaly that requires an explanation.
3
B. Income dependence of education.
Two aspects of education in children, educational expenditures and school dropouts have vast
data with regard to their dependence on the income of the parents [17-21]. Educational expenditures
vary directly with the income of the parents, well within the ambit of Engel’s curves of the concave
necessities and the convex luxuries and the in-betweens (Fig.2 Left) [20].
Fig 2: (Left) The three Engel curves of expenditure. (Right) The Engel curves were from a hierarchy of expenditures as in
Eq.3 replicating the phenomenology associated with the Engel curves, each expenditure being hyperbolic itself as a function
of income. The arrow represents increasing θ (cf. Eq. 1)
Fig. 3 depicts education and health (out of pocket expenditures) in comparison with cereals,
clothing and accommodation (rent) from the urban and rural households in the 50th round of National
Sample Survey Organization (NSSO) data of 1993-94 as an example since the pattern repeats over all
rounds [21]. A typical functional form often employed by economists and considered by them to be a
good approximation to the observed data is:
C= Ayθ
…Eq. 1.
Or Log C = log A+ θLogy where θ is referred to as the elasticity coefficient. Thus, θ<1 reflects
necessities and θ >1 reflects luxuries. These are represented in Fig. 2 Left with three curves; a convex, a
concave and intermediate, monotonic relationships.
4
150
MPCE
100
50
0
0
200
400
600
800
1000
Income
150
MPCE
100
50
0
0
500
1000
1500
2000
Income
Fig 3: 50th Round (1993-94) of NSSO published aggregate data. Top, Rural (68598 households), Bottom, urban (46148
households). Monthly per capita expenditures (MPCE) (in Indian rupees) of selected items: Closed circles, Cereals (major
food component), Inverted filled triangles, Rent, Filled squares, Health, Unfilled triangles, Education, Unfilled circles,
Clothing. Note that income in NSSO data always indicates the sum of all expenditures actually obtained in surveys and the
data over years shows an invariant pattern barring inflation.
While the log-log relationships have been adopted for the purpose of defining income
dependence (or elasticity) thus far, the methodology was found to be in reality highly unsatisfactory.
The foremost requirement in linear regression (on direct or transformed data) is that the residuals should
be random, i.e., there should be no residual pattern. By using runs test [22], a useful non-parametric test
that detects non-randomness of residuals, the log-log fits by regression were rejected uniformly though
the r2 itself was of the order of ≥ 0.9 in all NSSO data over some 20 odd rounds, both urban and
rural[23]! The data on cereals (food) clearly indicated a saturating behavior and we observed that when
fitted to a hyperbolic function (of the kind, y = ax/(b+x)), the residuals were invariably random unlike
the log-log fit. That is,
Exp 1 = Exp max1.I/(K1+I)
…Eq. 2.
5
Where Exp1 is the expenditure on the 1st commodity (i.e., with highest preference
reference such as cereals, the
staple source of carbohydrate), Expmax1 is the maximum value that this expenditure
nditure could take, K1 is the
inverse of preference for the commodity and I, the income. Given the consumption data as in Fig.
Fig 3, it
was shown that the parameters are estimated readily from the NSSO data for items such as cereals,
which account for the concave
cave form of Engel curve (necessities), quite adequately [20,23].
[2
It occurred to
us that if the preferences are ordered or hierarchic, the expenditure on the item of second preference
would also be hyperbolic except that the expenditure on the first item needs to be subtracted from the
total income to give the residual income Ir
Ir, against which is plotted the second expenditure preference
(Fig. 4 Bottom).. Given a sequence of commodities, this could be repeated in the order of increasing θ,
which approximately indicates K for each commodity. Thus
Exp1 = Expmax1.I/(K1+I)
Exp2 = Expmax2.Ir/(K2+Ir)
Exp3 = Expmax3.Ir’/(K3+Ir’)
where Ir= I- Exp1
where Ir’= I- (Exp1+ Exp2)
…Eq. 3.
…and so on. Fig 2 (Right) gives these theoretical profiles of expenditures constructed as a hierarchy and
we readily see that the relationships create all the three types of Engel curves. In other words, all
consumption is hyperbolic once we strip away the influence of hierarchy. The NSSO data in available
rounds, for both urban and rural, could be subjected to an algorithm that selects for a hierarchy based on
higher preference and accepts the models based on i. a better r2 and ii. random residuals in the runs test
[cf. 22].. The models chosen were the linear regression with and without log transformation and the
rectangular hyperbola by non-linear
linear regression.. The rectangular hyperbola succeeded in every round
over some 50 years [23]. This allowed us to define the hierarchica
hierarchically
lly ordered consumer basket for the
urban and the rural populations of India for the first time algorithmically using NSSO data without
additional assumptions. Educational
al expenditure is also best understood as a commodity in its income
dependence, amenablee to similar analys
analyses. The dropout rates (or
or its converse, attendance) which largely
depend on parental income would be the right choice to analyze the socioeconomic dependence of
education.
Fig 4: (Top).. Conventional models that partition income into various expenditures, considered at the same level such that
each relationship is defined by an elasticity coefficient directly related to income. (Bottom): A hierarchic model of
consumption. Each RI is the residual income after considering the expenditure for the commodit(y)ies of the higher
preference. The hierarchical model does not exclude some expenditures being at the same level as in top figure but the three
curves of Engel must follow the hierarchy wh
wherein
erein the residual income is obtained solving for Eq.3 serially.
6
It is imperative that we understand that ever since Engel determined in 1857, the three types of
expenditure in the area of econometrics, it continued without an insight/mechanism and remained a
brute empirical observation. The essence of the hyperbolic models of the kind, y= ax/(b+x) is commonly
used in other disciplines as well, such as enzyme kinetics in biochemistry and predator-prey
relationships in ecology as Holling type II model [24-26]. The proof of these models derives from the
simple fact that a micro-time constant associated with the material that is being processed exists…such
as the processing of each insect by a bird while eating or the making or breaking of a bond in catalysis
by an enzyme or a catalyst. Saturation occurs simply because the overall rate cannot even remotely
approximate the micro-time constant, much less supersede it. In the case of economics, clearly the
temporal distinction lies between perishables and durables which represent a continuous spectrum of
commodities rather than two distinct classes and it is surprising that such a formal distinction has not
been made except by us (Appendix A)[23,27-30].
C . The education model
The model was feasible due to the availability of published empirical data from Brazil [cf.13], tailored to
the construction of a serial model similar to Engel’s curves of consumption inductively (Fig 5). The
measurement relates to fractional attendance in each grade corrected for census, plotted as a function of
income class of the parents. The overall relationship, fij, between educational involvement over all the
grades and the parental income is nearly linear, (Fig 6, Left):
Aij = f ij (Ci, Ij…)
…Eq. 4.
Where Aij is the fractional attendance in the ith grade that also depends on the jth income class, Ci refers
to the grade (class of study) i, Ij refers to the income for jth class, measured in multiples (1-10) of
minimum wages (in Brazilian Reals, R$ then relevant in Brazil), such that,
i =12
∑A
ij
= f j ( I ) = mI j + c j
…Eq. 5.
i =1
Where m appears to be rather independent of j.
The fractional attendance follows a natural order such that higher the income, higher the fractional
attendance and shows concavity for the lower grades and convexity for the higher grades (classes)
(Fig.5, Bottom). Therefore A1j should be of the form
a1 .I j
…Eq. 6.
A1j =
b1 + I j
which is a concave function. Since {A1j} is but a subset of the entire data set {Aij}, the hierarchy of
preference renders, at i = 2, in a hypothetical set where there are only two grades of study considered,
a1 .I j
A2j = mIj + cj …Eq. 7a
b1 + I j
or, A2j = mIj + cj - A1j
... Eq. 7b
where grade 1 is concave and thereby grade 2 is convex such that the sum is linear, consistent with
reality when actually summed up over 12 grades. It simply means that the concave function becomes
progressively more convex with each consecutive grade of study, from 1-12, as a function of income.
Here Ai and Aj are data given while ai and bi are estimated. It is worth keeping in mind that this is a
unique and valuable national data corrected for census several decades ago! It also follows that, except
for the last grade that remains linear when the influence of grades i-1 are removed, i+1 becomes concave,
as we have seen earlier with commodities (Eq. 3).
7
0.8
0.6
0.4
4
6
0.0
10
8
8
6
Inco
de
2
0.2
Gra
Partic
ipatio
n
1.0
10
4
2
me
12
Participation
1
0
0
2
4
6
8
10
12
Income
Fig 5: (Top). Income (in units of minimal wages of 113 R$) vs participation upto 12 years of schooling in Brazil 1983,
corrected for census. (Bottom), same data plotted to indicate the similarity with Engel’s curves. The arrow indicates
increasing grades from 1 to 12.
8
12
6
8
Arbitrary units
Cumulative Fractional Attendance
7
10
6
5
4
3
4
2
2
1
0
0
0
2
4
6
8
10
12
0
2
4
6
8
10
12
Grade
Income class
Fig 6: (Left) The cumulative fractional attendance in the Brazilian education data was assessed for dependence of the
income class of the parents. The data across grades fitted a linear regression, y = 2.38 + 0.761x, r = 0.99, P < 0.001.
However, the residuals were not random such that the non-parametric Wald–Wolfowitz runs test revealed runs consistent
with the probability for non-randomness, P < 0.0005. (Right) The plot shows Vmax (closed circles), K (open circles) in
arbitrary units (axis on the left.) The values of K at corresponding grades could be fitted to a parabolic equation of the kind,
8
y = a.x (b – c.x) representing a shift in K at the critical grade, b at which the direction of the preference (inverse of K)
changes: specifically the data shows K = 0.35x (4.77 – 0.356x), the coefficient of correlation, r = 0.85, P <0.001, where x is
the grade, b approximates the 5th grade.
Methodology:
Studies on retention: The teachers in the State Board English medium school satisfied the
minimal educational criteria for their teaching commitments. The teachers declined to participate in
taking the tests on the grounds that they would not be familiar with subjects other than what they teach.
Permission could be obtained only for a limited number of students for testing and hence the choice of
students (N ≅ 30).
The teachers initially gave 600 multiple choice questions at 10 questions per subject (that they
actually taught) per year. These were with ≥ 20% errors. The test was reduced by the school to 280
questions with 4-6 questions per subject per year as the teachers felt that even a 5 hours test (of 600
questions) would be too taxing and with inadequate time. The final set of questions still had ~10%
errors, where the students were given the benefit. The actual test took 88 ± 14 min (i.e., <20 sec per
question) though time allotted was 240 min (~70 sec/question). The performance of 108 students in their
earlier Grade 8 was plotted for their scores (theory only) in school tests. A set of students falling into a
uniform distribution was handpicked exclusively based on a nearly constant interval between scores to
obtain 32 students, out of which 28 actually completed the test in two parts of 140 questions each.
Since scoring for retention should relate to the content learnt, we scanned the official textbooks
from the State Board to obtain a word count per page (though the random sampling tended to be biased
to more OCR(optical character recognition) software-friendly, as the books have number of figures and
unfilled spaces in tables. For Indian languages (Hindi and Marathi) we extrapolated the number of pages
from equal sized books in English and font size subjected to word count, further confirmed by actual
word count. The content of each subject was added for each year to get the total content per year in
words. The content was scaled over the years, considering the content in the first year as 1.0. This
permitted us to make a distinction between retention as the performance in questions for each grade
retrospectively, and learning, in arbitrary units, could be defined as a product of this performance
multiplied by the content for that grade.
Income dependence of education: Data of Costa Ribeiro was manually digitized from the
published work with less than 1% error and used. Data from National Sample Survey Organization for
the 50th round for selective consumables was plotted directly from the aggregate data [cf.21] confirmed
also by compiling and evaluating unit data [cf.29,30].
Results:
I. Study on retention.
A major problem in designing these studies was that the anticipated normal distribution of the
performance of the students would tend to confound the statistical relationships due to grouping around
the mean. We selected students based on equidistant performance in the latest available 8th grade school
scores to yield a near uniform (platykurtic) distribution, kurtosis being of the major concern (Fig. 7).
9
Fig 7: Frequency distribution of the 8th grade marks (filled circles, 67.8 ±10, n= 105, kurtosis, -0.165), students for the new
test selected to achieve a uniform distribution (unfilled circles, 68.8 ± 14, n= 28, kurtosis, -0.846) and the final result of the
new test (filled triangles, 68.8 ± 8.5, n=28, kurtosis, 0.648). Note that the 3 sets of results on average did not differ.
Fig. 8, Left shows the aggregate 8th grade scores plotted against the current test aggregate scores.
The relationship is significant even though 4 students among the 32 selected did not take the test. The
frequency distributions of the new test scores were considerably more leptokurtic than the sampled
uniform distribution based on 8th grade scores (Fig. 7), showing that the new test did not merely
replicate but measured something different than the conventional school tests [cf.8].
The critical observation was that the new test score showed an increase in variation at lower
scores i.e., marked heteroscedasticity (Fig.8, Left). This is important because at least 20% of the
children that would be labeled as poor performers in the class tests emerged with better scores indicating
comparable retentive ability as with the better performers, suggesting an unjust misclassification! Fig.8,
Right, reveals what was not anticipated. The score for 10th grade was not low and the Model III
regression, i.e., past performance, determined even the future 10th grade score (cf. Fig. 1, C- +x trend as
opposed to C – 10 trend, also see Fig.9)), raising distinct possibilities. Firstly, if the fall in scores shown
by Model III was due to lesser retention, i. it reflects a general tendency in education that the ability to
learn per se decreases, grave, if true; ii: they are taught little in 10th grade, largely spent in cramming,
preparative to Board exams, the merit of which itself questionable. Secondly, acquiring scores higher
than anticipated for 10th grade not yet taught could mean that a. they attend coaching classes (which is
true in majority of students), or, b. this test per se was very lenient as they were not anyway taught much
in 10th grade or even earlier. Reality could well be a combination of all these.
10
100
Grade 10 test scores, Recorded
1
Current test score,%
80
60
40
20
0.8
0.6
0.4
0.2
0
0
30
50
70
0.1
90
0.3
Grade 8 scores, total,%
0.5
0.7
0.9
Grade 10 test scores, Predicted
1.0
25
0.8
20
0.6
15
Content
Average Score
Fig 8: (Left): A comparison of performance by school tests and the current test (normalized to 100) for retention. Linear
regression of the 28 students who attended the test was by the method of least squares, y = 0.224x + 52.30, r=0.374, p <0.05.
Note that a subset of 5 students (~20%, highlighted as open circles) scored significantly higher than predicted by the school
tests in which they fared ~50% scores or less. Closed triangles, the students who did not appear for the test.
(Right): A comparison of test scores obtained for Std.10 with those predicted by individual regression analyses of 28 students
up to 9th grade, the 10th grade scores obtained by extrapolation of the regression equations. The predicted and recorded test
scores (both normalized to 1) were compared by linear regression obtained by the method of least squares, y= 0.435x +
0.244, r= 0.523, p<0.001.
0.4
0.2
5
0.0
0
0
2
4
6
8
10
0
Grade
14
12
10
8
6
4
2
0
2
4
4
6
8
10
12
6
8
10
Average test scores
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
Grade
16
Average test scores x Content
10
12
Grade
Fig 9: (Top, Left): Test scores averaged for the relevant subjects normalized to 1.0 (max) for Grade 1-10 among 28 students.
Vertical bars represent ± standard deviation among subjects for each grade/year. Regression line for all subjects, y = 0.943 0.040x, r= 0.889, P < 0.001, clearly in accordance with Model III (cf. Fig. 1).
11
(Top, Right): The aggregate content for all subjects was plotted over years/grades. (See materials and methods for
calculating content of subjects, total content of Grade 1 taken as 1.0). The relationship between content (not given) as well as
cumulative content (as shown here vs. grade) shows a highly significant relationship, y = 1.122x2 - 3.250x + 4.512, r =
0.998, P <0.0001.
(Bottom): Relationship between the average test scores (as a measure of retention, shown as filled squares, y =0.995 0.067x + 0.002x2, r = 0.898, P<0.001) and the test scores multiplied by the content for each year (as a measure of learning
for that year in arbitrary units, shown as filled circles, y =0.22 + 0.593x + 0.054x2, r =0.893, P < 0.001) as a function of the
grade studied, both fitted to a quadratic equation for ease of comparison.
Age and year of study dominate much of our thinking in education while the real important
variable is the content. Obtaining a proxy for the content (which is of course mostly written in words in
text books) was simply that we evaluated it as the sum of the number of words that can be recognized in
scans of pages of all textbooks and arrived at an aggregate number for each subject and year. The
average score in the current test shows a clear relationship to the year, including 10th grade indicating
that the Model III behavior was consistent with progressively compromised retention of the content (Fig
9, Top Left). It is important to note that the content increased significantly over the years such that the
content showed a clear second order relationship to the year of study (Fig 9, Top Right). While the
scores obtained per year reflect normalization, the quantum of learning needs to be a product of the
normalized score and the content for that year. As expected, we see that the quantum of learning
increases with each year while that fraction retained decreases dramatically (Fig. 9, Bottom). This is the
first report of its kind in educational studies to the best of our knowledge. How significant is the loss of
retention as in Model III?
Performance in education is considered to be highly susceptible to social variables and nature of
subjects [cf.8,31,32]. Table 1 shows that Model III behavior is common to Science, Mathematics,
History and English (the medium of instruction). The slope as well as its variance varied with the subject
(content and teaching, not further distinguishable in this study) such that the slopes were significantly
different between Science and English and Mathematics considered critical for STEM education. The
lesser sample size and larger variance did not allow a better assignment of a model for Geography, Hindi
(the national language) and Marathi (the local language).
12
Table 1: Applicability of Models for various subjects in the current test.
Subject
Grades
Science
1-10
Slope*
(S.E)
-0.06046
(0.0079)
r2
P
Model
0.877
<0.001
III
Mathematics 1-10
-0.02891
(0.0079)
0.623
<0.01
III
History
3-10
-0.05146
(0.0169)
0.607
<0.01
III
English
1-10
0.360
<.05
III
Geography
5-10
-0.02588
(0.0122)
-0.00472
(0.034)
0.004
NS
-
Hindi
5-10
0.00612
(0.0481)
0.004
NS
-
Marathi
5-10
-0.02883
(0.0237)
0.270
NS
-
Current performance is given for individual subjects over years, including the prospective 10th Grade. *Slope =average
score per student (max=1)/no. of grades, obtained by linear regression by the method of least squares. Slope for Science
differed significantly (p<0.05) with English and Mathematics only. Sign indicates models as in Fig.1. S.E., standard error of
the slope. N.S., Not significant.
We ranked the 8th grade scores into groups of three and assessed the slope across years. It is
generally presumed, often correctly, that the lower performers in schools are from the lower socioeconomic strata, a major concern and source of debate in a pluralistic society that is striving to be more
inclusive [33]. While the intercept (i.e., at grade 0) relates to school performance, the negative slopes
are quite independent of the school performance or gender, significance of which requires special
attention (Table 2).
13
Table 2: School scores for Grade VIII vs current test scores for groups of three students across all
subjects, for 1-10 years of study, based on linear regression by the method of least squares.
VIII Scores
Mean + S.D (n)
Slope
Intercept
175 ± 3.0 (3)
170 ± 1.5 (3)
161 ± 3.2 (3)
151 ± 1.7 (3)
142 ± 3.5 (3)
132 ± 4.0 (3)
121 ± 3.2 (3)
110 ± 6.2 (3)
88 ± 11 (4)
151 ± 25.4 (12 girls)
127 ±26.9(16 boys)
-0.0429
-0.0407
-0.0367
-0.0365
-0.0406
-0.048
-0.0427
-0.0439
-0.0417
-0.0411
-0.0419
1.007
0.994
0.921
0.951
0.973
0.927
0.940
0.889
0.926
0.966
0.932
r
0.865
0.887
0.799
0.794
0.860
0.9
0.902
0.834
0.950
0.897
0.902
P
<0.001
<0.001
<0.01
<0.01
<0.001
<0.001
<0.001
<0.001
<0.001
<0.001
<0.001
The slope was not related to the school score by linear regression, y= 4*10-5x - 0.047, r= 0.3535, N.S. On the
other hand, the intercept was significantly related to school scores y= 9.3*10-4x -0.058, r = 0.719, P <0.01.
Number of students of each score group, n, is given in parentheses. The slopes for each group of three were
indistinguishable from each other, though the slope itself for each group was significant, shown by the r and P
values. Bottom two sets of data relate to gender non-selectivity.
The essential difference in this new test compared with the usual school examinations was both
in the nature of the examination and the performance in relationship to the year-wise content. Since the
test was given by the teachers themselves with questions that they felt most representative of each year
and subject, there was no influence of any theory of pedagogy or ideology in the test per se. Higher
variance in humanities and languages (which cannot be ignored in the name of STEM education), which
also contribute to Model III, particularly need to be traced to problems in teaching methods and content.
That this test, with all its limitations and no external intervention, could give a robust attribute,
the slope, α, raises two distinct possibilities to ponder. Either it is a measure of the limits to mass
learning or it is a measure of the current practices of learning and offers a completely novel and unique
metric for the competence of systems of education, i.e., the students, teachers and the curriculum by
appropriate emphases in analyses.
II. Income dependence of school drop outs.
a. International data.
Despite huge efforts worldwide, mass literacy programs are limited by parental incomes in
developing countries [cf.19]. Though primary education is mostly free, even with subsidies tried in
many countries, (e.g., cash, food provided for the school going children etc.) the dropout rates continue
to be high [cf.13,19].
Where does one find socioeconomic influences on education besides the usual school tests which
are difficult to model with any level of accuracy or insight? An examination of UNESCO or World
Bank data [cf.18,19] on completion of primary schooling as a function gross domestic product or per
capita income has enormous scatter wherein it is difficult to discern meaningful relationships. We came
across an important Brazilian report with data that was amenable to precise modeling, though decades
later.
14
b. Brazilian data:
Fig.6 Left shows the cumulative fractional attendance (sum of attendance in each grade as a
fraction of the census such that the sum cannot exceed the total number of grades) (cf.13), plotted as a
function of income for the Brazilian population. While the relationship is monotonic, clearly the
fractional attendance shows a plateau midway in income classes. More importantly, fractional
attendance and parental income are reasonable proxies for each other such that we could emulate our
earlier analyses of Engel’s curves.
Based on the assumption that some expenditure is incurred for each grade, the fractional
attendance representing a measure of that cost, we subjected the data to analysis similar to consumption
data of NSSO [23]. The results were striking in that while the maximum attendance did not exhibit much
change, the K value for each grade increased and subsequently decreased, exhibiting a parabolic
relationship at high level of significance (P< 0.001) (Fig.6, Right). These findings were consistent with
the following observations: i. Education, the world over, is income related; ii. This income dependence
is quite similar to other consumption data; iii. the income dependence of schooling follows the natural
hierarchy of grades exceptionally well; and iv. The explanation of a biphasic relationship in K values
for education is easily understood…at lower incomes and lower grades with the end far in sight,
education is seen more as a burden, a situation that changes midway as the end is in sight and the
attitudes correspondingly change. Thus, the hierarchical analyses capture the social nuances in the
decisions that parents indulge in, rather well. This needs to be contrasted with heroic attempts thus far
made to capture the reason for high drop-out rates world over, wherein the detail attempted lost the
essence, viz., parental income [34, 35]. When proxy variables were used, the dependence of dropout
rates on parental income was lost [36]. Thus, the decision for dropout rate at the lower school level,
tragic as it is, is parental, while the decision at the collegiate level is an individual adult decision [cf.16],
unfortunate in terms of wastage of national resources.
Discussion:
There is nothing that preempts large, complex social systems from exhibiting highly tractable
and even simple behaviours [37]. Education is one among many such universal social processes. Its
economic dependence is manifest in the early years of schooling wherein it follows the rigorous
contours of a hierarchical econometric theory, highly consistent with our previous work [cf. 23, 27-30].
The influence of socioeconomic conditions percolates into higher education in conventional schooling
and performance which involves cramming. The mistrust in the conventional grading system based on
year-end examinations is justified to the extent that it tends to mis-categorize those who learn and retain
and yet are not capable of cramming. Periodic intensive cramming evidently requires a social setting that
allows the student to minimize distractions in the short term for the purpose of examinations.
School dropouts:
It is clear that the results reported here regarding retention in learning require large scale
confirmation in and outside the country, possible only by a strict adherence to the methodological
innovations introduced here. On the other hand, school dropouts and attainment of universal literacy
(even if reduced to the simplest form of reading and writing) represent a well documented and yet, a
highly remediable situation, despite the economic constraints. It demands a social momentum as the
prime mover as demonstrated in Kerala in this country even without gross economic development [3840]. Since the major focus on school dropouts shifts to rural areas, it is prudent to consider that the
economic upliftment requires equal attention besides a spread of awareness; the huge social cost of
dropouts (which far exceeds that of any intervention) requires that it must be minimized by any and all
means.
15
Fall in Retention:
The major result here relates to the revelation of a metric, α, of fall in retention that is
independent of the socio-economic status and gender. What to do to ameliorate the situation is evident
from the results already. Since the fall, α, varies with the subject (i.e. content), it defines the focal point
for development in the design and delivery of the content and the examination system as well. What
makes a subject be forgotten readily? Clearly that learning fails to relate to our lives, i.e., the knowledge
that is ‘bookish’ as opposed to ‘practical’ has little to enforce its retention. We see that the retention
remains highest in the first three years of schooling, since these basics are a prerequisite for all the
subsequent years, while the subsequent years do not offer such a rigorous continuity.
Learning in ‘silos’:
The irrelevance of the subjects taught has two foci: firstly each subject, as evidenced by the
textbooks, is independent of all others, i.e., in silos; secondly, so do teachers restrict their involvement to
little beyond their own teaching in a specified subject. How do we expect the students to correlate,
integrate and develop a homogenous view of their learning independent of the content and the teachers?
The usability of the subject has two considerations. One relates to extent of usage across subjects such as
Science in Mathematics or English in Science and the other relates to across years, whose continuity can
be assessed by concept maps and other tools [41-43]. A major revision of content should aim at, say,
more than 30% overlap between subjects while the teachers themselves be proficient in more than 2-3
subjects. The aim, which does not exclude teachers, is obviously rendering α to 900.
Testing as an adjunct to actual learning:
Evaluation by examinations itself is a practical way to enhance retention, to the extent that we
need to consider examinations not as something to be feared but as a useful adjunct to learning [44].
Then the examination system has to change from the year-end torture to self-help ad libitum by students
themselves, or, as assisted by teachers, largely to appreciate what they have NOT learnt. What if the
content itself is measured by the number of non-repetitive class of questions that it can generate, rather
than the number of pages or even the weight of the book? That needs to be supplemented by efforts that
use and reuse the earlier content in the later years, supported by clearly continuous concept maps
creating a demand for recall and utilization. The student has to see the relevance of what is learnt within
the learning process itself and/or in his life at school and home. That relates well to the normal practices
in professional education, e.g., medicine or engineering.
These were also deeply embedded notions in education in India for millennia as evidenced by
Panchatantra [cf.2], which even taught value systems as practicable entities. The ultimate example of the
finest mixture of poetry and language and mathematics is embodied in Lilavati that Bhaskara II wrote
for his daughter [45]. We could identify many such texts in English and other languages and even
creative writing of new interdisciplinary texts is eminently feasible.
All this is possible once we accept, unconditionally, that evaluation is an integral form of
education. We had on several occasions discussed the relevant matters with several teachers, parents and
students as to their perceptions, though not measured. Dilution in the methods of evaluation has become
indispensible; the school administration is answerable to the performance statistics rather than
performance per se [46]. The only way is to promote most, if not all, students as the schools need to
cope with large number of failing students [47]. Justifications include the ‘psychological stress’[48]
imposed on the students by examinations, a source of anxiety for parents! This argues strongly for a
fresh look at examinations as a learning device and using modern technology to reduce the tedium for
both students and teachers. Laptops and automation are better suited for testing and not as a substitute to
teaching.
16
Universality of problems of pedagogy: the general case
Our results are unlikely to be specific for the country, India or Brazil, and share the universal
concern for the poor state of education, with each country visualizing and battling its own problems.
This in itself points out the inadequacy of the existing theories of pedagogy in visualizing a common
metric. Barring war-torn, oppressive and disaster-struck areas [49], if the decline in retention is
independent of peoples and intrinsic to the current process of education itself, a major effort to examine
the alternatives will be worthwhile. The teachers who formulate examinations may not themselves fare
well in the same examinations across subjects. If the teachers take the same tests as the students and the
performance of the two mutually match and outrank the other pairs of teachers and students between
schools… that would be a realistic assessment of the teaching process. Learning necessarily implies
retention. The current results do not even follow the direction predicted by models (e.g. Model II) of the
much maligned rote learning, which should really be viewed as pre-principle learning, mandatory if
learning has to be effective. While rote learning has a place in schooling, rote teaching has none. The
resulting reorientation in educational methodologies and reforms will not be small in eliminating
learning/teaching of subjects in silos (the content mostly suffers from this ‘isolation’ of subject matter),
which appears to be a major source of disorder in achieving coherence in teaching!
The decline in learning could indicate limits to learning itself (i.e., of principles) at the mass
level, a worrisome but a real prospect, since it also sets practical limits to what can be achieved. Any
approach that makes education prohibitively expensive (e.g., due to unregulated privatization and
market forces) can be evaluated for its true effectiveness and curtailed if not justified. We need to extend
the current focused and yet limited study, primarily empirical rather than theory based, to wider national
and international levels that include students, teachers and curriculum(and possibly value systems), a
feat within easy reach in these days of automation and big data handling capabilities, given the will [50].
The need for good data and competent handling by modeling:
Focused, well designed and small studies that are aware of the pitfalls of the experimental
designs give an unambiguous direction to our understanding but fail in ascertaining the universally
applicable prime movers in inducting social change. Large data is meaningful only when curated well,
as is characteristic of the Indian NSSO data and the Brazilian data on education. It is critical to combine
both wherever possible. Our studies on consumption data on Indian populations led to an index for
poverty which did not require an arbitrary poverty line (which has led to much political and social unrest
in the recent years in India)[cf.23,28-30]. Such a claim would be considered an overstatement if it were
not realized that Engel curves represent a. a hierarchy of expenditures and b. these share an underlying
mechanism of the time constant associated with the commodity responsible for the saturating behaviour
as also seen in catalysis and ecology of predation. Considering how the Engel curves for consumption
and Costa Ribeiro’s data on income dependence of completion of schooling remained unresolved for
several decades, the importance of rigorous modeling of empirical phenomena is self evident. The plea
by Iaonnidis [51] for statistical veracity in medical research was found to be very valid in education and
econometrics in our own studies, which shared the basis of much of the novel results reported here.
Education being the only commodity that is naturally ordered (in the sense that attendance in grades
1-12, is obligatorily sequential), it also confirms similar work from our group on the consumption
preferences among rural and urban Indian populations. Consumption requires to be ordered since its
ordinal nature is not given a priori. The results obtained on econometrics and econometrics of education
differ from most of the current work, not because of any preconceived models, but by strict adherence
to the best practices, e.g., i. avoiding transformations on data and ii. directly handling data with nonlinear regression methodology, iii. actually and assiduously confirming at every stage that the residuals
were indeed random by non-parametric tests such as runs test and, since correlation does not guarantee
causality, iv. to arrive at derivable models based on ‘mechanisms’ in the physical sense to assess the
validity of the conclusions. It is amazing that a log-log relationship is still adopted by econometricians
17
without ever validating whether the residuals were acceptably random! Our hierarchical model could get
rid of the arbitrary and contentious poverty line and led to a commodity-specific poverty index without
any axiomatic approaches that defy independent proof [cf. 23,28-30]. The hierarchy inherent to
consumption and even education as a commodity has not been visualized in previous studies
[e.g.,52,53].
Lastly, the studies yielded totally unforeseen dividends in that the affinity (1/K) for investing on
education goes downhill initially but actually begins to increase midway through schooling as the end is
in sight. Even if primary education is considered ‘free’, the hidden costs of that education that the family
incurs which includes the attention paid to the children determines the success of continuing education
towards completion of schooling. Value systems play a major role which includes the decision processes
at home. Unfortunately, each commodity, education, health etc., represent domains of expenditure and
are analyzed independent of each other, while a family makes a decision on each expenditure
representing the sum total of the needs and aspirations. We see that in rural India, education plays a
secondary role to health expenditures while the urban India sees them with equal emphasis.
Clearly, the triad of learning, teaching and retention could be assessed independent of
socioeconomic circumstances. We also see that socioeconomic circumstances could be assessed by
different parameters such as dropout rates quite independently. More importantly, the strongly empirical
testing approaches developed here can be duplicated anywhere in the world and can be assessed for the
veracity of such a conclusion, which it should be. Viewing testing, as in conventional sense, as a
necessary evil should cease and testing which does not require cramming but helps identify to the
student, what is not known, should become an integral part in learning. The importance of a measure of
learning per se, i.e., α, independent of socioeconomic circumstances, is novel and much desired. The
results from large data bases are emphatic enough and have been proven to be valid in all the years from
NSSO data. The models so arrived at speak volubly for their validation in other landscapes of learning
since pedagogy is one area of human social interest undimmed over millennia and yet fraught with much
conflict, theoretical, value based, social and religious and carries its own baggage.
The need for a change in set paradigms.
The difficulty in reforms lies in their execution in a community that does not have the clarity of
what the prospects are, as with STEM education. The isolationist policy of cherry-picking students after
school, based on conventional evaluations, into extremely selective institutes (like IITs etc., in India,
prevalent in rest of the world as well) cannot substitute for a national strategy for STEM education [54]
since it needs to be mass-based, aiming at the creators as well as users of that knowledge. At the same
time, the income-dependent [cf. 12,18], high dropout rates (> 30% in India) and grossly inadequate
learning at the mass level do not indicate any success of public policies (cf. studies by Pratham [55] and
elsewhere [56], followed by preliminary reports of baseline tests by the several Indian states [57-59] as
with the rest of the world [14]). Insights that facilitate causally relevant intervention for students are few.
Evaluation of teachers is equally dismal [60]. Percentage of teachers that pass the National teacher
eligibility test (TET) at the middle school level in India varies from 0.6% (2012) to 12% (2015) [61,62].
The third critical component, the curriculum design itself, is with hardly any metric or feedback [cf. 15,
63].The ability to formulate varied and yet unambiguous questions offers a good method to assess both
the teachers and the curriculum, when quantified.
Through the social pressures in the community, the relatively less aware may find an isolationist
approach to subjects and careers preferable; it is so not because of any virtue associated with
compartmentalized learning but due to the intellectual myopia characteristic of ingrained ignorance of
preceding generations that learnt mostly in silos. It appears very plausible that reforms in teaching,
testing and curricula that aim at a progressive erasure of learning in silos, are most likely to achieve an
improvement in α towards 90o. The practice of elimination of students not capable of mere cramming
alone would amply justify these efforts.
18
Acknowledgements: The experiment was not financially supported by any external agency and was
primarily executed by the authors themselves, supported by teachers and students in an English medium
school in an urban Indian setting.
Individual contributions: Sitaramam formulated the problem and designed the experiment and
completed the analysis, modeling and finalizing the paper. Iyer initially raised the question of how the
teacher training in Bachelor of Education courses match the teaching requirements and subsequent
performance of both teachers and students. She helped in screening the question banks and preparing the
school data for analyses and the manuscript. The first author (VS) acknowledges Julia R.S. da Silva who
performed the analyses of Brazilian Education data and thanks Dr. Paulo dos S. Rodrigues for help in
collating the Brazilian data, while VS was on a Sabbatical at Universidade Federal do Rio de Janeiro.
There are no conflicting interests.
19
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Appendix A: A comparison of the three uses of the hyperbolic function in different domains of
biochemistry, ecology in comparison with the present use in econometrics/education. Stepwise
derivation of the proof.
1. What saturates ?
Enzymology (catalysis)
Predator-prey
relationships*
Econometrics
Total time for catalysis,
Total time spent in
predation, Tt
Total time for occupancy
for a commodity, Tt
Tt
2. What has not
Time not involved in actual
catalysis, Ts
Time spent in the search
for the prey, Ts
Time to spare, Ts
3. The microevent
Time to handle one
substrate(turnover time) Th
= 1/Tn ,
Time to handle one prey,
Th
Time to handle one unit of
commodity, Th
Tn = turnover number (or
Vmax/Et, as conventional)
4. Number of events
No. of substrate molecules
catalyzed per unit time, Na
No. of prey captured, Na
Number of units of
commodity purchased, Na
5. Density
Concentration of substrate,
S
Prey density, C
Expenditure total
(=income), Expt
Na ∝ S.Ts
Na ∝ C.Ts
Na ∝ Expt.Ts
i.e., Na =a .Ts. S
i.e.,Na=a.C.Ts
= a.Expt.Ts
6. Balance equation
Tt=Ts+NaTh
Tt = Ts +Na.Th
Tt=Ts+NaTh
7. Saturable function
Rate of catalysis
Rate of consumption
Rate of consumption
= Na/Tt
= Na /Tt
= Na /Tt
= aS/(1+a.S.Th)
= a.C/(1+a.C.Th)
= a.Expt / (1+a.Expt.Th)
= (1/Th).S/((1/a.Th) +S
=[(1/Th).C] / [(1/aTh)+C]
=(1/Th).Expt /((1/(a.Th))+
Expt),
=Tn.S/(Tn.a)+S
where (1/Th) is the
maximum rate of
consumption.
*Holling Type 2.
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